The number of real solutions of the equation \((x^2 - 15x + 55)^{x^2 - 5x + 6} = 1\) is __________.
Let \( ABC \) be a triangle right-angled at \( B \) with \( AB = BC = 18 \). The area of the largest rectangle that can be inscribed in this triangle and has \( B \) as one of the vertices is _____________.
A fruit seller has oranges, apples, and bananas in the ratio 3:6:7. If the number of oranges is a multiple of both 5 and 6, then the minimum number of fruits the seller has is ______________.
The number of pairs \( (x, y) \) of integers satisfying the inequality \( |x - 5| + |y - 5| \leq 6 \) is ____________.
The price of a chocolate is increased by \( x% \) and then reduced by \( x% \). The new price is 96.76% of the original price. Then \( x \) is:
The following table shows the number of employees and their median age in eight companies located in a district. \[ \begin{array}{|l|l|l|} \hline \textbf{Company} & \textbf{Number of Employees} & \textbf{Median Age} \\ \hline A & 32 & 24 \\ B & 28 & 30 \\ C & 43 & 39 \\ D & 39 & 45 \\ E & 35 & 49 \\ F & 29 & 54 \\ G & 23 & 59 \\ H & 16 & 63 \\ \hline \end{array} \] It is known that the age of all employees are integers. It is also known that the age of every employee in A is strictly less than the age of every employee in B, the age of every employee in B is strictly less than the age of every employee in C, and the age of every employee in G is strictly less than the age of every employee in H. The median age of an employee across the eight companies is …………..
The following table shows the number of employees and their median age in eight companies located in a district. \[ \begin{array}{|l|l|l|} \hline \textbf{Company} & \textbf{Number of Employees} & \textbf{Median Age} \\ \hline A & 32 & 24 \\ B & 28 & 30 \\ C & 43 & 39 \\ D & 39 & 45 \\ E & 35 & 49 \\ F & 29 & 54 \\ G & 23 & 59 \\ H & 16 & 63 \\ \hline \end{array} \] It is known that the age of all employees are integers. It is also known that the age of every employee in A is strictly less than the age of every employee in B, the age of every employee in B is strictly less than the age of every employee in C, and the age of every employee in G is strictly less than the age of every employee in H. In company F, the lowest possible sum of the ages of all employees is ………….
If \(4 \log_2 x - 4 \log_3 x - 16 x + 68 = 0\), then \(x - 2\) equals ………...
Person A borrows rupees 4000 from another person B for a duration of 4 year rupee He borrows a portion of it at 3% simple interest per annum, while the rest at 4% simple interest per annum. If B gets rupees 520 as total interest, then the amount A borrowed at 3% per annum in rupee is …………….
The number of triangles with integer sides and with perimeter 15 is ………… .
